The Bantams were in desperate need of a victory after two successive defeats and they started brightly against their defensive minded opponents.
Stuart McCall's promotion hopefuls dominated the early play and it seemed only a matter of time before they scored but the home fans had to be patient as the chances came and went in a scrappy first period.
City striker Michael Boulding saw two headers go straight at keeper Jonny Brain and after 35 minutes Furman set up skipper Graeme Lee only for the Bradford centre-half to fire wide from the edge of the box.
Town offered little going forward with Gareth Evans playing up front on his own and City were left cursing their luck six minutes before the break.
More good work by Furman opened up the Macclesfield defence but Steve Jones' first time shot from 18 yards crashed against the crossbar and away to safety.
It was more of the same after the interval with the visitors barely mustering an attack and City took a deserved lead on the hour mark.
The energetic Furman was again at the heart of the move and after playing a smart one-two with Boulding he kept his cool to beat Brain with a low shot from 16 yards.
Three minutes later Brain did well to keep out a fierce drive from the hard working Nicky Law and Macclesfield never threatened to grab an equaliser.
Brain again saved from Law in the closing stages and although City substitute Peter Thorne had the ball in the net in injury time his effort was ruled out for offside.